APPENDIX 685 



point of greatest frequency of the theoretical distribution, of which the 

 given distribution is a sample. As a disadvantage, it should be mentioned 

 that it is somewhat difficult to determine the theoretical mode accurately. 

 It may also be pointed out that for a very irregular group of figures the 

 mode is practically useless. Its great service is to characterize a type, and 

 with a very irregular group of figures the existence of a type is not mani- 

 fest ; indeed, a type may not exist. 



The median. If all the variates are arranged in serial order, the value 

 corresponding to the middle variate is called the median of the population. 

 Thus, if we should speak of the wages of a thousand and one laborers, we 

 should mean by the median the wages of the middlemost of these men 

 when they are arranged in serial order with respect to wages ; that is, if 

 five hundred received less than $1.72, and five hundred received more than 

 $1.72, we should say that $1.72 is the median wage. The median has the 

 great advantage that it can be easily determined. Very large and very 

 small values do not affect it. It is only a question of being above or below 

 the middle in an arrangement. Its great disadvantages are that it may be 

 totally removed from the type and that it gives no special importance to 

 extreme values. 



Averages of whatever kind are designed to exhibit the main features of 

 a population by means of a few well-chosen numbers. We have seen that 

 the particular average selected depends upon the purpose we have in view. 

 Now, if this purpose is merely one of comparison between two similar 

 groups, then almost any kind of an average will do. The ordinary and the 

 weighted means have been used for the mo^t part in this work, but con- 

 ditions may easily arise where some other average is more suitable. Bowley 

 has well stated the following as the characteristics of a good and suitable 

 average : " If there is a type, it shows it ; it gives due influence to extreme 

 cases ; it is not easily affected by errors, or much displaced by slight 

 alterations in the system of calculations ; and it is easily calculated." 



It is often useful to give more than one average in order to describe a 

 population ; for the relative positions of the mean, the mode, and the median 

 give a good deal more information about the distribution of a population 

 than any single average can give. For a great many distributions Pearson 

 has found an approximate relation to exist between the mean, the mode, 

 and the median. This relation is 



theoretical mode = mean 3 (mean median). 



It is, of course, possible to form fictitious frequency distributions for 

 which this relation does not hold, but it is important as indicating what 

 nature, in general, provides. 



The use of averages for representing what is often spoken of as 

 the "true value" can be better discussed in the section devoted to the 

 probable error. 



